Answer:
The measures of angles of triangle MNP are
Step-by-step explanation:
The picture of the question in the attached figure
step 1
Find the measure of arcs AB, BC and AC
we know that
The inscribed angle is half that of the arc it comprises.
so
[tex]\gamma=\frac{1}{2}[arc\ AB] ----> arc\ AB=2\gamma\\\alpha=\frac{1}{2}[arc\ BC] ----> arc\ BC=2\alpha\\\beta=\frac{1}{2}[arc\ AC] ----> arc\ AC=2\beta[/tex]
step 2
Find the measure of angle M
we know that
The measurement of the outer angle is the semi-difference of the arcs it encompasses.
[tex]M=\frac{1}{2}[arc\ AB+arc\ BC-arc\ AC][/tex]
substitute
[tex]M=\frac{1}{2}[2\gamma+2\alpha-2\beta]\\M=[\gamma+\alpha-\beta][/tex]
step 3
Find the measure of angle N
we know that
The measurement of the outer angle is the semi-difference of the arcs it encompasses.
[tex]N=\frac{1}{2}[arc\ AC+arc\ BC-arc\ AB][/tex]
substitute
[tex]N=\frac{1}{2}[2\beta+2\alpha-2\gamma]\\N=[\beta+\alpha-\gamma][/tex]
step 4
Find the measure of angle P
we know that
The measurement of the outer angle is the semi-difference of the arcs it encompasses.
[tex]P=\frac{1}{2}[arc\ AC+arc\ AB-arc\ BC][/tex]
substitute
[tex]P=\frac{1}{2}[2\beta+2\gamma-2\alpha]\\P=[\beta+\gamma-\alpha][/tex]
